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A comparison between Collocation and Galerkin Isogeometric approximation of acoustic wave problems.
No full textIsogeometric collocation methods combine the high smoothness of NURBS basis functions with the low computational cost of collocation methods, generating sparser stiffness and mass matrices than the ones generated by isogeometric Galerkin methods. Our previous work investigated the approximation of 2D acoustic wave problems with proper absorbing boundary conditions by Galerkin IGA methods in space and Newmark’s explicit schemes in time (IGA-Gal-New). In this talk, we extend our study to IGA collocation explicit and implicit approximations (IGA-Col-New). A detailed numerical study on both Cartesian and NURBS domains illustrate the stability and convergence properties of the two Isogeometric Newmark methods with respect to the IGA and Newmark discretization’s parameters. The experimental results show that the stability thresholds of the methods depend linearly on h and inversely on p, confirming that the proposed IGA-Col-New method retains the good convergence and stability properties of standard IGA-Gal-New and Spectral Element discretizations of acoustic problems.
Moreover, a detailed comparison of convergence errors, CPU time, and matrix sparsity patterns show that IGA-Col-New often outperforms IGA-Gal-New, in particular in the case of maximal regularity k = p - 1 and for increasing NURBS degree p. Some numerical results on the spectral properties of the IGA-Col-New matrices are also mentioned
Isogeometric methods for acoustic waves with absorbing boundary conditions
Full text linkIn recent years there has been an increasing attention to high order simulation of acoustic and elastic wave
propagation. In this presentation we consider the Galerkin and Collocation Isogeometric approximation
of the acoustic wave equation with absorbing boundary conditions in cartesian and curvilinear 2D
regions, while the time discretization is based on explicit or implicit Newmark schemes.
Since both the IGA Galerkin and Collocation mass matrices are not diagonal, the main dierence between
explicit and implicit IGA Newmark schemes is related to the stability bounds for the time step, rather
than to the solution of the linear systems arising at each temporal instant.
In this respect we briefly illustrate some stability estimates both for the semidiscrete and fully discrete
schemes that are only partially based on proven results, due to the lack of theoretical estimates regarding
eigenvalues and conditioning of the mass and stiffness IGA matrices.
Furthermore, we present a detailed numerical study on the properties of the IGA methods as concerns
stability thresholds, convergence errors, accuracy, and spectral properties of the IGA matrices
varying the polynomial degree p, mesh size h, regularity k, and time step.
Finally, we focus on two meaningful examples in the framework of wave propagation simulations: a test
problem with an oscillatory exact solution having increasing wave number, and the propagation of one or
two interfering Ricker wavelets.
Numerical results show that the IGA Collocation method retains the convergence and stability properties
of IGA Galerkin. Moreover, IGA Collocation is in general less accurate when we adopt the same choices of
discretization parameters. On the other hand, regarding the computational cost and the amount of memory
required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the
IGA Galerkin method, especially in the case of maximal regularity k = p - 1 with increasing NURBS
degree p
Isogeometric approximations of the scalar wave equation
Full text linkIn recent years several there has been an increasing attention to high order simulations
of acoustic and elastic wave propagation. While our previous works focused on approxi-
mations based on spectral and spectral elements methods, we then extended our study to
Isogeometric (IGA) methods that allow not only the standard p- and hp- re nement of hp-
nite elements and spectral elements, where p is the polynomial degree of the C0 piecewise
polynomial basis functions, but also a novel k- refi nement where the global regularity k of
the IGA basis functions is increased proportionally to the degree p, up to the maximal IGA
regularity k = p - 1.
In this presentation we consider the numerical approximations of the acoustic wave equation with absorbing boundary conditions, that are introduced in order to simulate wave
propagation in in nite domains, by truncating the original unbounded region into a nite
one. The spatial discretization is based on IGA Galerkin and Collocation in cartesian
and curvilinear 2D regions, while the time discretization is based on explicit or implicit
Newmark schemes. We illustrate a detailed experimental study of the two IGA methods
with regard to spectral properties of the IGA mass and stiffness matrices, stability, accuracy and convergence of the IGA schemes with respect to all the discretization
parameters, namely the local polynomial degree p, regularity k, mesh size h, and the time
step size of the Newmark schemes. Finally we show some preliminar numerical results
on the application of an additive overlapping Schwarz preconditioner to both IGA Galerkin
and Collocation approximations, testing its performance with GMRES or preconditioned
conjugate gradients iterative methods
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